[R-sig-ME] glmmTMB: variance-covariance matrix parameterization

[Sophia Kyriakou]

Hi all,

Does anyone know what is the parameterization that glmmTMB uses for the
variance-covariance matrix in the case of generelized linear mixed models?

I can tell that when fitting a generalized linear mixed model with a random
intercept, where the random effects are normally distributed with zero mean
and variance sigma^2, glmmTMB estimates theta = log(sigma) and then returns
sigma^2.

But what is the parameterization for example in the 2 x 2 random-slopes
case?
Let the 2 x 2 variance-covariance matrix have elements ( sigma^2_{1},
sigma_{12}, sigma_{12}, sigma^2_{2} ).
glmmTMB returns three theta parameters (theta_1, theta_2, theta_3), where
theta_1 = log(sigma_1)
theta_2 = log(sigma_2)
but I don't know what the relationship between theta_3 and sigma_{12} (or
the correlation rho) is.

I know that I can extract thetas using fitTMB$sdr$par.fixed and sigmas
using matrix(unlist(VarCorr(fitTMB)),2,2), where fitTMB is the model fitted
via glmmTMB, but I would like to understand the parameterization used.

Thank you in advance.

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